Optics 020 · Weighing a lone black hole

Gravitational Lens Telescope

How can light bent around an unseen mass reveal how heavy it is? From the images it makes of a star behind it. The mass bends the star's light into two images, one each side of a circle called the Einstein ring, whose angular size grows with the square root of the mass. The images' spacing and brightness, with the distances, give the mass. In 2025 Hubble weighed a lone, dark black hole of about 7 solar masses from the shift these images give the star's light.

Drag a black hole across the line of sight to a star. The star brightens and splits into two images, a ring or arcs, and their spacing gives the mass.

How much brighter the star looks $\text{—}$

Separation of the two images $\text{—}$

Shift of the star’s combined light $\text{—}$

Mass worked out from the images $\text{—}$

Einstein radius $\text{—}$

Days to cross one Einstein radius $\text{—}$

Difference in parallax of lens and star $\text{—}$

Lens’s distance from the line of sight $\text{—}$

Edge points traced for this record $\text{—}$

Optics 020

How heavy is the black hole that bends this star’s light?

Days from closest approach

t

-60\,\mathrm{d}

Mass of the black hole

M

7.15\,M_{\odot}

Closest approach to the line of sight

u_{0}

0.00271\,\theta_{\mathrm{E}}

Distance to the black hole

D_{\mathrm{L}}

1.52\,\mathrm{kpc}

Radius of the background star

R_{\ast}

2.88\,R_{\odot}

No telescope split this event’s two images, 11 mas apart. Hubble measured the shift of their combined light instead, which also grows with the Einstein radius.

Model assumptions

The lens is a single point mass and the light is bent by the weak-field result of general relativity. The event follows OGLE-2011-BLG-0462 as Sahu et al. (2025) measured it: an Einstein radius of 5.496 mas crossed in 273.1 days, so that lens and star move apart at 7.35 mas a year, a closest approach of 0.00271 Einstein radii, and a lens of 7.15 solar masses 1.52 kpc away in front of a star 7.1 kpc away. The star stays at that distance for every setting. Its default radius of 2.88 solar radii gives the 0.0038 mas diameter found by Sahu et al. (2022). The lens moves in a straight line: the Earth’s yearly motion, which made the parallax measurable, is left out, so the light curve is symmetric about closest approach. The star is a uniform disc without limb darkening, and no light from other stars is blended in. The images are found by tracing 720 points round the star’s edge through the lens equation. Their areas give the brightening, which is exact for a uniform disc, and the light-weighted centre of both images gives the shift. The mass from the images uses each image’s light centre and the two brightnesses. It is exact for a point-like star and for a complete ring, whose radius comes from the outer and inner edges, but reads low when the images are wide arcs. In the side scene, distances across the line of sight are drawn about 28 million times those along it, and the stars and the black hole are enlarged; the sky image uses true angles.

Physics tutorial

Weigh a black hole that gives no light

BackgroundOn 2 June 2011 the OGLE and MOA surveys both announced that a star towards the centre of the Galaxy was brightening. It peaked at 369 times its normal brightness, and the rise and fall took months. Nothing could be seen in front of it. Kailash Sahu and colleagues then imaged the star with the Hubble Space Telescope for eleven years and found that it had also moved slightly on the sky. In 2025 they put the unseen lens at 7.15 ± 0.83 solar masses, 1.52 kpc away: the first black hole found alone, without a companion star.

Why it mattersA black hole on its own gives no light and pulls on nothing we can watch. But gravity bends the light of any star that happens to lie behind it, by an amount set by the mass. If the bending can be measured, the black hole can be weighed without ever being seen.

Start with the essentials

Focus question: How do the brightening and the images of a background star tell how heavy an invisible lens is?
One-sentence intuition: Light from the star passes the black hole on both sides and reaches the telescope as two images, one outside and one inside a circle of angular radius $\theta_{\mathrm{E}}$ , the Einstein radius. For OGLE-2011-BLG-0462 it is 5.5 mas, about the width of a coin seen from 700 km. The product of the two images’ distances from the lens is always $\theta_{\mathrm{E}}^{2}$ , and $\theta_{\mathrm{E}}^{2}$ grows in proportion to the mass, so the images and the distances together give the mass.

Core mathematical model

The Einstein radius grows with the square root of the mass

\begin{gathered}\theta_{\mathrm{E}}=\sqrt{\kappa M\pi_{\mathrm{LS}}},\quad\kappa=\frac{4G}{c^{2}\,\mathrm{AU}}=8.144\,\frac{\mathrm{mas}}{M_{\odot}}\\\pi_{\mathrm{LS}}=1\,\mathrm{AU}\left(\frac{1}{D_{\mathrm{L}}}-\frac{1}{D_{\mathrm{S}}}\right)\end{gathered}

A lens of mass $M$ at distance $D_{\mathrm{L}}$ in front of a star at $D_{\mathrm{S}}$ bends the star’s light into a ring of angular radius $\theta_{\mathrm{E}}$ when the three line up. $\pi_{\mathrm{LS}}$ is the difference in parallax of lens and star. Turned round, $M=\theta_{\mathrm{E}}^{2}/(\kappa\pi_{\mathrm{LS}})$ : Sahu et al. (2025) found the mass from $\theta_{\mathrm{E}}=5.496\,\mathrm{mas}$ and the parallax measured in the light curve.

Every star behind a point mass has two images

\begin{gathered}\beta=\theta-\frac{\theta_{\mathrm{E}}^{2}}{\theta}\\\theta_{\pm}=\frac{\beta\pm\sqrt{\beta^{2}+4\theta_{\mathrm{E}}^{2}}}{2},\quad\theta_{+}\theta_{-}=-\theta_{\mathrm{E}}^{2}\end{gathered}

A star at angle $\beta$ from the lens appears at the angles $\theta$ that solve the lens equation. There are always two, $\theta_{+}$ outside the Einstein ring on the star’s side and $\theta_{-}$ inside it on the far side. When the star lies right behind the lens, every direction round it works and the images merge into a ring.

Brightness and the brightness ratio

\begin{gathered}A=\frac{u^{2}+2}{u\sqrt{u^{2}+4}},\quad u=\frac{\beta}{\theta_{\mathrm{E}}}\\r=\frac{A_{+}}{\lvert A_{-}\rvert}=\left(\frac{\theta_{+}}{\theta_{-}}\right)^{4}\;\Rightarrow\;\theta_{\mathrm{E}}=\frac{\Delta\theta}{r^{1/4}+r^{-1/4}}\end{gathered}

Bending stretches each image without changing its surface brightness, so a larger image is brighter. $A$ is the total brightening of a point-like star at $u$ Einstein radii from the lens; it grows without limit as $u$ falls. The two images’ brightness ratio $r$ and separation $\Delta\theta$ are enough to find $\theta_{\mathrm{E}}$ without knowing where the dark lens is.

The star’s light appears to move

\delta=\frac{u}{u^{2}+2}\,\theta_{\mathrm{E}},\qquad\delta_{\max}=\frac{\theta_{\mathrm{E}}}{2\sqrt{2}}\ \text{at}\ u=\sqrt{2}

When the images are too close to split, their combined light still sits off the star’s true place by $\delta$ (Sahu et al. 2022, Eq. 6). It is largest at $u=\sqrt{2}$ , long after the brightening has faded, and it is proportional to $\theta_{\mathrm{E}}$ . Hubble measured this shift.

Common difficulties

The black hole is not seen at all

Typical misconceptionThe images show the black hole itself, outlined by bent light.

Better mental modelAll the light in the images comes from the background star. The black hole only bends it; its own size, about 21 km for 7 solar masses, is far too small to show, and Sahu et al. found no light from the lens.

Brighter does not mean closer to us

Typical misconceptionThe star brightens because the lens brings it closer or focuses it like a magnifying glass at one point.

Better mental modelThe star stays where it is. Its images are stretched round the lens, and since bending keeps surface brightness the same, the larger images carry more light. The brightening depends only on how far the lens is from the line of sight, in Einstein radii.

The light curve alone does not give the mass

Typical misconceptionHow long the event lasts tells how heavy the lens is.

Better mental modelThe 273-day timescale is the Einstein radius divided by how fast lens and star move apart on the sky, and that speed is not known in advance. The size of the images, or the shift of their light, gives the Einstein radius itself; the distances or the parallax then give the mass.

The two-image formula needs a small star

Typical misconceptionThe separation and brightness ratio give the exact mass whatever the star.

Better mental modelThe formula is exact for a point-like star. When the star is a sizeable fraction of the Einstein radius, the images become arcs whose light centres move, and the formula drifts: 9 % low in the arcs preset. A full ring gives the radius again from its two edges.

Run the experiment

01
See the two images
Keep the default preset, Two images of one star.
What to observe: The sky image shows the star as an outer image beyond its true place and a fainter inner image on the other side of the black hole, 11.0 mas apart. Together they are 4.63 times brighter than the star alone, and the separation and brightness ratio give 7.15 solar masses.
02
Drag the black hole across the line of sight
Drag the black hole sideways in the scene, or move the date slider.
What to observe: Closer to the line, both images brighten and the separation shrinks towards the diameter of the Einstein ring, 11.0 mas. The light curve and the two plots of shift and separation move their cursor with you.
03
Watch the peak
Load Closest approach: 370 times brighter.
What to observe: The star is 370 times brighter, close to the 369 recorded in 2011. The images sit almost on the Einstein ring, on opposite sides, and their combined light is back at the star’s true place: the shift is only 0.007 mas.
04
Put a big star behind it
Load A big star behind it: Einstein ring.
What to observe: With a star 1000 times the Sun’s radius right behind the black hole, the images join into a ring 11.0 mas across, 16.8 times brighter. A larger star is brightened less at its peak, because its parts lie at different distances from the lens.
05
Move the big star slightly off
Load Slightly off: two arcs.
What to observe: The ring breaks into two arcs. Their light centres are no longer where a point-like star’s images would be, so the two-image formula reads 6.49 solar masses, 9 % low.
06
Try a neutron star
Load A neutron star: smaller ring, shorter event.
What to observe: At 1.4 solar masses the Einstein ring is 2.43 mas in radius, the images are 5.0 mas apart and the event lasts 121 days across one Einstein radius. In 2022 one team put the lens at 1.6 to 4.4 solar masses, most likely a neutron star or a light black hole; later Hubble images favoured a black hole of about 7.

Start with the essentials

Focus question: How does a star’s finite size change a planetary microlensing signal?
One-sentence intuition: Gravity preserves surface brightness. Integrate the same limb-darkened image area to obtain both the visible image and its total flux.

Core mathematical model

A binary bends the same light

\zeta=z-\sum_{j=1}^{2}\frac{m_j}{\bar z-a_j}

Angles use the Einstein radius of total lens mass; the binary axis is real and its center of mass is zero.

Area supplies the light

A_*={\int I(|\zeta(z)-\zeta_c|)\,d^2z\over\pi\rho^2\bar I}

The image-plane radiance cells supply both the linear exported image and the source flux. Display contrast is enhanced separately.

Other stars dilute the change

A_{\rm ap}=1+f(A_*-1)

The source fraction is measured before lensing. The separate foreground disk stays constant and contributes to the same aperture image and flux.

Common difficulties

A caustic is a diagnostic

Typical misconceptionThe caustic is a luminous object beside the lens.

Better mental modelIt is a curve in the source plane where ideal point-source images become critical. The luminous physical images appear in the separate sky view.

A short event need not be a light lens

Typical misconceptionMeasuring the event duration directly weighs the lens.

Better mental modelDuration also depends on geometry and relative motion. A physical mass requires additional angular-scale and parallax information or explicitly stated population assumptions.

Run the experiment

01
Drag through the feature
Pause Planetary brightening. Drag the green source handle and watch the changing images and current point on the light curve.
What to observe: The curve and image update from one finite-source integration; the source-plane caustic is shown separately as a diagnostic.
02
Give the star more area
Keep the path fixed and increase the angular source radius. Compare the width and height of the small feature.
What to observe: Different parts of the stellar disk cross the caustic at different times, smoothing a point-source spike.
03
Add constant light
Restore the radius, then reduce the source fraction. Compare the cyan source curve with the gold total curve.
What to observe: Foreground light dilutes the feature without changing the source’s gravitational magnification.

Gravitational Lens Telescope

Weigh a black hole that gives no light

Core mathematical model

The Einstein radius grows with the square root of the mass

Every star behind a point mass has two images

Brightness and the brightness ratio

The star’s light appears to move

Common difficulties

The black hole is not seen at all

Brighter does not mean closer to us

The light curve alone does not give the mass

The two-image formula needs a small star

Run the experiment

See the two images

Drag the black hole across the line of sight

Watch the peak

Put a big star behind it

Move the big star slightly off

Try a neutron star

Follow the images before reading the curve

Core mathematical model

A binary bends the same light

Area supplies the light

Other stars dilute the change

Common difficulties

A caustic is a diagnostic

A short event need not be a light lens

Run the experiment

Drag through the feature

Give the star more area

Add constant light

Related experiments and articles

From bent starlight to a planetary light curve

Gravity leaves a trace